Normal not implies normal-extensible automorphism-invariant in finite
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a finite group, every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., normal-extensible automorphism-invariant subgroup)
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Statement with symbols
- Normal-extensible not implies normal
- Normal-extensible not implies extensible
- Normal-extensible not implies inner
- Normal not implies semi-strongly potentially relatively characteristic
- Potentially characteristic not implies semi-strongly potentially relatively characteristic
- Normal not implies semi-strongly potentially characteristic, normal not implies strongly potentially characteristic
- Potentially characteristic not implies semi-strongly potentially characteristic, potentially characteristic not implies strongly potentially characteristic
- Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible
- Automorphism group of direct power of simple non-abelian group equals wreath product of automorphism group and symmetric group
Example of the dihedral group
Let be the dihedral group of order eight. Then, every automorphism of fixes every element of the center of , and also, the inner automorphism group of is maximal in the automorphism group of . Thus, by fact (1), every automorphism of is normal-extensible.
However, there is an automorphism of that interchanges the two normal Klein four-subgroups. Thus, these two normal subgroups are not invariant under this automorphism, and hence, we have an automorphism of that is normal-extensible but not normal.
Equivalently, the Klein four-subgroups are examples of normal subgroups that are not normal-extensible automorphism-invariant.
Example involving a simple complete group
Let be a simple complete group. In other words, is a centerless simple group such that every automorphism of is inner. Let . By fact (2), the automorphism group of is the wreath product of with the symmetric group of degree two, which has , the inner automorphism group, as a subgroup of index two. Moreover, is centerless. Thus, by fact (1), we get that every automorphism of is normal-extensible.
However, the coordinate exchange automorphism of , that interchanges the two copies of , is not a normal automorphism because it interchanges these two normal subgroups. Thus, we have an example of a normal-extensible automorphism that is not normal.
Equivalently, either of the direct factors is an example of a normal subgroup that is not normal-extensible automorphism-invariant.